By mimicking brain-like cognition and exploiting parallelism, hyperdimensional computing (HDC) classifiers have been emerging as a lightweight framework to achieve efficient on-device inference. Nonetheless, they have two fundamental drawbacks, heuristic training process and ultra-high dimension, which result in sub-optimal inference accuracy and large model sizes beyond the capability of tiny devices with stringent resource constraints. In this paper, we address these fundamental drawbacks and propose a low-dimensional computing (LDC) alternative. Specifically, by mapping our LDC classifier into an equivalent neural network, we optimize our model using a principled training approach. Most importantly, we can improve the inference accuracy while successfully reducing the ultra-high dimension of existing HDC models by orders of magnitude (e.g., 8000 vs. 4/64). We run experiments to evaluate our LDC classifier by considering different datasets for inference on tiny devices, and also implement different models on an FPGA platform for acceleration. The results highlight that our LDC classifier offers an overwhelming advantage over the existing brain-inspired HDC models and is particularly suitable for inference on tiny devices.